Notes: `(r + 3s)^6` Use the Binomial Theorem to expand and simplify the expression.

We have to use the binomial theorem to expand the expression, so use the formula:

$\displaystyle{{\left({a}+{b}\right)}}^{{n}}={\sum_{{{k}={0}}}^{{n}}}{\left(\matrix{{n}\\{k}}\right)}{{a}}^{{{n}-{k}}}{{b}}^{{k}}$

$\displaystyle\therefore{{\left({r}+{3}{s}\right)}}^{{6}}={\left(\matrix{{6}\\{0}}\right)}{{r}}^{{{6}-{0}}}{{\left({3}{s}\right)}}^{{0}}+{\left(\matrix{{6}\\{1}}\right)}{{r}}^{{{6}-{1}}}{{\left({3}{s}\right)}}^{{1}}+{\left(\matrix{{6}\\{2}}\right)}{{r}}^{{{6}-{2}}}{{\left({3}{s}\right)}}^{{2}}+{\left(\matrix{{6}\\{3}}\right)}{{r}}^{{{6}-{3}}}{{\left({3}{s}\right)}}^{{3}}+{\left(\matrix{{6}\\{4}}\right)}{{r}}^{{{6}-{4}}}{{\left({3}{s}\right)}}^{{4}}+{\left(\matrix{{6}\\{5}}\right)}{{r}}^{{{6}-{5}}}{{\left({3}{s}\right)}}^{{5}}+{\left(\matrix{{6}\\{6}}\right)}{{r}}^{{{6}-{6}}}{{\left({3}{s}\right)}}^{{6}}$

$\displaystyle={{r}}^{{6}}+\frac{{{6}!}}{{{1}!{\left({6}-{1}\right)}!}}{{r}}^{{5}}{\left({3}{s}\right)}+\frac{{{6}!}}{{{2}!{\left({6}-{2}\right)}!}}{{r}}^{{4}}{\left({9}{{s}}^{{2}}\right)}+\frac{{{6}!}}{{{3}!{\left({6}-{3}\right)}!}}{{r}}^{{3}}{\left({27}{{s}}^{{3}}\right)}+\frac{{{6}!}}{{{4}!{\left({6}-{4}\right)}!}}{{r}}^{{2}}{\left({81}{{s}}^{{4}}\right)}+\frac{{{6}!}}{{{5}!{\left({6}-{5}\right)}!}}{{r}}^{{1}}{\left({243}{{s}}^{{5}}\right)}+{729}{{s}}^{{6}}$

$\displaystyle={{r}}^{{6}}+\frac{{{6}\cdot{5}!}}{{{5}!}}{\left({3}{{r}}^{{5}}{s}\right)}+\frac{{{6}\cdot{5}\cdot{4}!}}{{{2}\cdot{4}!}}{\left({9}{{r}}^{{4}}{{s}}^{{2}}\right)}+\frac{{{6}\cdot{5}\cdot{4}\cdot{3}!}}{{{3}!\cdot{3}\cdot{2}\cdot{1}}}{\left({27}{{r}}^{{3}}{{s}}^{{3}}\right)}+\frac{{{6}\cdot{5}\cdot{4}!}}{{{4}!\cdot{2}\cdot{1}}}{\left({81}{{r}}^{{2}}{{s}}^{{4}}\right)}+\frac{{{6}\cdot{5}!}}{{{5}!}}{\left({243}{r}{{s}}^{{5}}+{729}{{s}}^{{6}}\right.}$

$\displaystyle={{r}}^{{6}}+{18}{{r}}^{{5}}{s}+{135}{{r}}^{{4}}{{s}}^{{2}}+{540}{{r}}^{{3}}{{s}}^{{3}}+{1215}{{r}}^{{2}}{{s}}^{{4}}+{1458}{r}{{s}}^{{5}}+{729}{{s}}^{{6}}$

Notes

Saturday, May 1, 2010

$\displaystyle{{\left({r}+{3}{s}\right)}}^{{6}}$ Use the Binomial Theorem to expand and simplify the expression.

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